We report experimental results on the joint statistics of the Lagrangian acceleration and velocity in highly turbulent flows. The acceleration was measured up to a microscale Reynolds number R(lambda)=690 using high speed silicon strip detectors from high energy physics. The acceleration variance was observed to be strongly dependent on the velocity, following a Heisenberg-Yaglom-like u(9/2) increase. However, the shape of the probability density functions of the acceleration component conditioned on the same component of the velocity when normalized by the acceleration variance was observed to be independent of velocity and to coincide with the unconditional probability density function of the acceleration components. This observation imposes a strong mathematical constraint on the possible functional form of the acceleration probability distribution function.